Finally, to reinforce our analysis, we evaluate the forecasting performance of the three CAViaR competing models using the scoring function in (REF ). Specifically, at each time \(t\) , and for the specified level \(\tau \) , we define by \(S_{MAL_{t}}^{(j)}(\tau )\)  the scoring function associated to model \(j\) , and denote the difference between the scoring function of model \(i\)  and model \(j\)  by \(\Delta _{MAL, t}^{(i,j)} =S_{MAL_{t}}^{(i)}(\tau ) - S_{MAL_{t}}^{(j)}(\tau )\) , where \(i,j=1,2,3\) . We then test for the null hypothesis that \(\mathbb {E}[\Delta _{MAL, t}^{(i,j)}]=0\)  against \(\mathbb {E}[\Delta _{MAL, t}^{(i,j)}]<0\)  using the [1]} test, for all the pairs of models \(i\)  and \(j\) .
If the null hypothesis is rejected, then forecasts delivered by model \(i\)  are more accurate than those of model \(j\) , and therefore model \(i\)  is preferable than model \(j\) . The results of the test, together with the corresponding p-values, are reported in Table REF . The table clearly shows that the CAViaR-AS specification outperforms both the CAViaR-IG and CAViaR-SAV models at all the three quantile levels and for both the adopted ES formulations of constant multiplicative factor (Panel A) and AR dynamics (Panel B). Therefore, in order to select the best performing model between the two CAViaR-AS in Panels A and B, we apply again the [1]} test on the two competing CAViaR-AS specifications. The results are reported in Table REF  and suggest that the CAViaR-AS model with the ES specified as a constant multiple of the VaR provides the most accurate predictions at all the three quantile levels. This is in line with [3]}, who also find that the same specification not only produces the smallest losses, but it also delivers the most accurate predictions than all the other competing CAViaR dynamicsTo further justify this choice, we also compare the CAViaR-AS model with the Quantile AutoRegression of [4]}. Specifically, we estimate the regression model of [5]} using the lagged returns (at lag 1) of each asset as covariates. Comparing these two models would allow one to evaluate the potential contribution of assuming a CAViaR specification in the quantile dynamics. According to the coverage tests and the scoring functions defined in (REF ) and (REF ), we still find a better performance of the CAViaR-AS specifications.. These results corroborate the fact that accounting for asymmetries in the autoregressive process of a given quantile improves the model's forecasting ability (see, e.g., [6]}, [7]}, [8]} and [9]}).
